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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,0,42,21,74]))
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pari:[g,chi] = znchar(Mod(407,5160))
\(\chi_{5160}(263,\cdot)\)
\(\chi_{5160}(287,\cdot)\)
\(\chi_{5160}(407,\cdot)\)
\(\chi_{5160}(503,\cdot)\)
\(\chi_{5160}(743,\cdot)\)
\(\chi_{5160}(863,\cdot)\)
\(\chi_{5160}(1007,\cdot)\)
\(\chi_{5160}(1103,\cdot)\)
\(\chi_{5160}(1223,\cdot)\)
\(\chi_{5160}(1367,\cdot)\)
\(\chi_{5160}(1703,\cdot)\)
\(\chi_{5160}(2183,\cdot)\)
\(\chi_{5160}(2327,\cdot)\)
\(\chi_{5160}(2567,\cdot)\)
\(\chi_{5160}(2807,\cdot)\)
\(\chi_{5160}(2927,\cdot)\)
\(\chi_{5160}(3167,\cdot)\)
\(\chi_{5160}(3287,\cdot)\)
\(\chi_{5160}(3383,\cdot)\)
\(\chi_{5160}(3503,\cdot)\)
\(\chi_{5160}(3767,\cdot)\)
\(\chi_{5160}(4103,\cdot)\)
\(\chi_{5160}(4247,\cdot)\)
\(\chi_{5160}(4463,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((3871,2581,1721,3097,4561)\) → \((-1,1,-1,i,e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5160 }(407, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{14}\right)\) |
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sage:chi.jacobi_sum(n)
